Theorem idcncfg

Description: The identity function is a continuous function on CC . (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	idcncfg.a	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	idcncfg.b	⊢ ( 𝜑 → 𝐵 ⊆ ℂ )
Assertion	idcncfg	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝑥 ) ∈ ( 𝐴 –cn→ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		idcncfg.a	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		idcncfg.b	⊢ ( 𝜑 → 𝐵 ⊆ ℂ )
3		cncfmptid	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℂ ) → ( 𝑥 ∈ 𝐴 ↦ 𝑥 ) ∈ ( 𝐴 –cn→ 𝐵 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝑥 ) ∈ ( 𝐴 –cn→ 𝐵 ) )